How to move vertices while modeling?
I'm not quite sure how the mesh has got like this, but essentially I think the mesh has been scaled (in object mode, I think) so it has zero thickness (in the object panel the 'z' scale value is '0') which I think is why you can't move anything in edit mode. Any movement is being 'squashed' back to zero.

To fix this you can go to object mode, select the mesh and press Ctrl+A> Scale. This 'applies' the scale, resetting the scale values to '0' but leaving the mesh appearance unchanged.

This question may help, though applying scale can be a complex subject:

Why are the longer edges being beveled more?